Example 65.4 Known G and R

This animal breeding example from Henderson (1984, p. 48) considers multiple traits. The data are artificial and consist of measurements of two traits on three animals, but
the second trait of the third animal is missing. Assuming an additive genetic model, you can use PROC MIXED to predict the
breeding value of both traits on all three animals and also to predict the second trait of the third animal. The data are
as follows:

The preceding data are in the dense representation for a GDATA=
data set. You can also construct a data set with the sparse representation by using Row, Col, and Value variables, although this would require 21 observations instead of 6 for this example.

The MMEQ
and MMEQSOL
options request the mixed model equations and their solution. The variables Trait and Animal are classification variables, and Trait defines the entire matrix for the fixed-effects portion of the model, since the intercept is omitted with the NOINT
option. The fixed-effects solution vector and predicted values are also requested by using the S
and OUTP=
options, respectively.

The random effect Trait*Animal leads to a matrix with six columns, the first five corresponding to the identity matrix and the last consisting of 0s. An unstructured
matrix is specified by using the TYPE=UN
option, and it is read into PROC MIXED from a SAS data set by using the GDATA=
G specification. The G
and GI
options request the display of and , respectively. The S
option requests that the random-effects solution vector be displayed.

Note that the preceding matrix is block diagonal if the data are sorted by animals. The REPEATED
statement exploits this fact by requesting to have unstructured 22 blocks corresponding to animals, which are the subjects. The R
and RI
options request that the estimated 22 blocks for the first animal and its inverse be displayed. The PARMS
statement lists the parameters of this 22 matrix. Note that the parameters from are not specified in the PARMS
statement because they have already been assigned by using the GDATA=
option in the RANDOM
statement. The NOITER
option prevents PROC MIXED from computing residual (restricted) maximum likelihood estimates; instead, the known values are
used for inferences.

The "Unstructured" covariance structure (Output 65.4.1) applies to both and here. The levels of Trait and Animal have been specified correctly.

Output 65.4.1: Model and Class Level Information

The Mixed Procedure

Model Information

Data Set

WORK.H

Dependent Variable

Y

Covariance Structure

Unstructured

Subject Effect

Animal

Estimation Method

REML

Residual Variance Method

None

Fixed Effects SE Method

Model-Based

Degrees of Freedom Method

Containment

Class Level Information

Class

Levels

Values

Trait

2

1 2

Animal

3

1 2 3

The three covariance parameters indicated in Output 65.4.2 correspond to those from the matrix. Those from are considered fixed and known because of the GDATA=
option.

Output 65.4.2: Model Dimensions and Number of Observations

Dimensions

Covariance Parameters

3

Columns in X

2

Columns in Z

6

Subjects

1

Max Obs per Subject

5

Number of Observations

Number of Observations Read

6

Number of Observations Used

5

Number of Observations Not Used

1

Because starting values for the covariance parameters are specified in the PARMS
statement, the MIXED procedure prints the residual (restricted) log likelihood at the starting values. Because of the NOITER
option in the PARMS
statement, this is also the final log likelihood in this analysis (Output 65.4.3).

Output 65.4.3: REML Log Likelihood

Parameter Search

CovP1

CovP2

CovP3

Res Log Like

-2 Res Log Like

4.0000

1.0000

5.0000

-7.3731

14.7463

The block of corresponding to the first animal and the inverse of this block are shown in Output 65.4.4.

The table of covariance parameter estimates in Output 65.4.7 displays only the parameters in . Because of the GDATA=
option in the RANDOM
statement, the G-side parameters do not participate in the parameter estimation process. Because of the NOITER
option in the PARMS
statement, however, the R-side parameters in this output are identical to their starting values.

Output 65.4.7: R-Side Covariance Parameters

Covariance Parameter Estimates

Cov Parm

Subject

Estimate

UN(1,1)

Animal

4.0000

UN(2,1)

Animal

1.0000

UN(2,2)

Animal

5.0000

The coefficients of the mixed model equations in Output 65.4.8 agree with Henderson (1984, p. 55). Recall from Output 65.4.1 that there are 2 columns in and 6 columns in . The first 8 columns of the mixed model equations correspond to the and components. Column 9 represents the Y border.

Output 65.4.8: Mixed Model Equations with Y Border

Mixed Model Equations

Row

Effect

Trait

Animal

Col1

Col2

Col3

Col4

Col5

Col6

Col7

Col8

Col9

1

Trait

1

0.7763

-0.1053

0.2632

0.2632

0.2500

-0.05263

-0.05263

4.6974

2

Trait

2

-0.1053

0.4211

-0.05263

-0.05263

0.2105

0.2105

2.2105

3

Trait*Animal

1

1

0.2632

-0.05263

2.7632

-1.0000

-1.0000

-1.7193

0.6667

0.6667

1.1053

4

Trait*Animal

1

2

0.2632

-0.05263

-1.0000

2.2632

0.6667

-1.3860

1.8421

5

Trait*Animal

1

3

0.2500

-1.0000

2.2500

0.6667

-1.3333

1.7500

6

Trait*Animal

2

1

-0.05263

0.2105

-1.7193

0.6667

0.6667

1.8772

-0.6667

-0.6667

1.5789

7

Trait*Animal

2

2

-0.05263

0.2105

0.6667

-1.3860

-0.6667

1.5439

0.6316

8

Trait*Animal

2

3

0.6667

-1.3333

-0.6667

1.3333

The solution to the mixed model equations also matches that given by Henderson (1984, p. 55). After solving the augmented mixed model equations, you can find the solutions for fixed and random effects in the
last column (Output 65.4.9).

Output 65.4.9: Solutions of the Mixed Model Equations with Y Border

Mixed Model Equations Solution

Row

Effect

Trait

Animal

Col1

Col2

Col3

Col4

Col5

Col6

Col7

Col8

Col9

1

Trait

1

2.5508

1.5685

-1.3047

-1.1775

-1.1701

-1.3002

-1.1821

-1.1678

6.9909

2

Trait

2

1.5685

4.5539

-1.4112

-1.3534

-0.9410

-2.1592

-2.1055

-1.3149

6.9959

3

Trait*Animal

1

1

-1.3047

-1.4112

1.8282

1.0652

1.0206

1.8010

1.0925

1.0070

0.05450

4

Trait*Animal

1

2

-1.1775

-1.3534

1.0652

1.7589

0.7085

1.0900

1.7341

0.7209

-0.04955

5

Trait*Animal

1

3

-1.1701

-0.9410

1.0206

0.7085

1.7812

1.0095

0.7197

1.7756

0.02230

6

Trait*Animal

2

1

-1.3002

-2.1592

1.8010

1.0900

1.0095

2.7518

1.6392

1.4849

0.2651

7

Trait*Animal

2

2

-1.1821

-2.1055

1.0925

1.7341

0.7197

1.6392

2.6874

0.9930

-0.2601

8

Trait*Animal

2

3

-1.1678

-1.3149

1.0070

0.7209

1.7756

1.4849

0.9930

2.7645

0.1276

The solutions for the fixed and random effects in Output 65.4.10 correspond to the last column in Output 65.4.9. Note that the standard errors for the fixed effects and the prediction standard errors for the random effects are the square
root values of the diagonal entries in the solution of the mixed model equations (Output 65.4.9).

Output 65.4.10: Solutions for Fixed and Random Effects

Solution for Fixed Effects

Effect

Trait

Estimate

StandardError

DF

t Value

Pr > |t|

Trait

1

6.9909

1.5971

3

4.38

0.0221

Trait

2

6.9959

2.1340

3

3.28

0.0465

Solution for Random Effects

Effect

Trait

Animal

Estimate

Std Err Pred

DF

t Value

Pr > |t|

Trait*Animal

1

1

0.05450

1.3521

0

0.04

.

Trait*Animal

1

2

-0.04955

1.3262

0

-0.04

.

Trait*Animal

1

3

0.02230

1.3346

0

0.02

.

Trait*Animal

2

1

0.2651

1.6589

0

0.16

.

Trait*Animal

2

2

-0.2601

1.6393

0

-0.16

.

Trait*Animal

2

3

0.1276

1.6627

0

0.08

.

The estimates for the two traits are nearly identical, but the standard error of the second trait is larger because of the
missing observation.

The Estimate column in the "Solution for Random Effects" table lists the best linear unbiased predictions (BLUPs) of the breeding
values of both traits for all three animals. The p-values are missing because the default containment method for computing degrees of freedom results in zero degrees of freedom
for the random effects parameter tests.

Output 65.4.11: Significance Test Comparing Traits

Type 3 Tests of Fixed Effects

Effect

Num DF

Den DF

F Value

Pr > F

Trait

2

3

10.59

0.0437

The two estimated traits are significantly different from zero at the 5% level (Output 65.4.11).

Output 65.4.12 displays the predicted values of the observations based on the trait and breeding value estimates—that is, the fixed and
random effects.

Output 65.4.12: Predicted Observations

Obs

Trait

Animal

Y

Pred

StdErrPred

DF

Alpha

Lower

Upper

Resid

1

1

1

6

7.04542

1.33027

0

0.05

.

.

-1.04542

2

1

2

8

6.94137

1.39806

0

0.05

.

.

1.05863

3

1

3

7

7.01321

1.41129

0

0.05

.

.

-0.01321

4

2

1

9

7.26094

1.72839

0

0.05

.

.

1.73906

5

2

2

5

6.73576

1.74077

0

0.05

.

.

-1.73576

6

2

3

.

7.12015

2.99088

0

0.05

.

.

.

The predicted values are not the predictions of future records in the sense that they do not contain a component corresponding
to a new observational error. See Henderson (1984) for information about predicting future records. The Lower and Upper columns usually contain confidence limits for the predicted values; they are missing here because the random-effects parameter
degrees of freedom equals 0.